Combinatorics of $(m,n)$-Word Lattices

TL;DR

We address the combinatorics of the $(m,n)$-word lattices $\mathbf{W}(m,n)$, a quotient of the $m$-lighted $n$-shade rotation lattices arising from Hochschild polytopes. The main approach proves that for all $m,n\ge 0$, $\mathbf{W}(m,n)$ is extremal and interval-constructable by interval doublings. We determine cardinalities, canonical join representations, a Chapoton-style $H$-triangle, and the Galois graph, establishing a rich lattice-theoretic structure for this two-parameter family. These results generalize Hochschild lattice properties to a broader class of lattices, linking to generalized Hochschild polytopes and their combinatorics.

Abstract

We study the $(m,n)$-word lattices recently introduced by V. Pilaud and D. Poliakova in their study of generalized Hochschild polytopes. We prove that these lattices are extremal and constructable by interval doublings. Moreover, we describe further combinatorial properties of these lattices, such as their cardinality, their canonical join representations and their Galois graphs.

Figures (2)

• Figure 1: The doubling procedure that produces $\mathbf{W}(2,3)$ from $\mathbf{W}(2,2)$. Along each arrow, the highlighted interval is doubled.
• Figure 2: The Galois graph of $\mathbf{W}(2,3)$.

Theorems & Definitions (47)

• Theorem 1.1
• Proposition 2.1: day92doubling
• Lemma 2.2: freese95free
• Theorem 2.3: day79characterizations
• Theorem 2.4: thomas19rowmotion
• Definition 3.1: pilaud23hochschild
• Theorem 3.2: pilaud23hochschild*Corollary 82
• Lemma 4.1
• proof
• Lemma 4.2